26 Hours left, and we just hit another stretch goal!
almost 4 years ago
– Fri, May 15, 2020 at 01:01:31 PM
We're almost down to the last day, and with a last rush of support in these closing hours, we've just hit a stretch goal!
This is an exciting one. It's email support for the cards. One of the problems with getting new games/toys/objects is that you use them for a while and then kind of forget about them. And when it's something as versatile as these visual flash cards, you might find you use them in just one way, but don't explore other options for what they can do.
Since we hit this stretch goal, I'll be trying a new experiment to counter some of that inertia. After you receive your cards, I'll email out a new question, provocation, or idea of exploration once a week, for twelve weeks. Ideally, this should give you new ideas for how to use the cards to explore multiplication more deeply using the cards as a tool.
I think it'll be cool to try. Thanks for getting us to this point! More mini-games, and if we really surge, an app are still ahead as the final two stretch goals!
Final 48 Hours!
almost 4 years ago
– Thu, May 14, 2020 at 09:11:57 AM
We're in the final stretch of the campaign. Now is the moment to let your friends and colleagues know about it! There are still 3 stretch goals to hit, and with a strong final push, we could get them all!
I've been using the cards to write about multiplication in other places too. They're incredibly useful. Check out my blog post, just published today, about understanding multiplication.
Or watch this video on the same topic!
Update on the level 1 deck
about 4 years ago
– Thu, May 07, 2020 at 05:18:28 PM
We've been designing and redesigning various components of the deck. One pretty major improvement we decided on is a revamp of the "level 1" deck. Instead of using the ten frames, we're using the "subitizing" model, using easy to see arrangements of dots in similarly arranged groups. Here's a peek at what they look like.
When you're choosing representations to use, there are always choices, and always trade-offs. I was drawn to the ten-frame approach originally, but I feel this version does a better job of introducing the conceptual meaning of multiplication. It's also more distinct from the array model that comes next.
I'm happy we keep finding ways to make the cards better :-)
We've just got a week left in the campaign! If you know anyone who might like these cards, now's the time for them to jump onboard! Thanks, as always, for your support!
A meditation on commutativity of multiplication
about 4 years ago
– Wed, May 06, 2020 at 10:57:59 AM
We're in the final stretch of the campaign now! The last ten days will determine just how successful the campaign is. Keep spreading the word, and hopefully we'll get to 4x of our campaign target, hit all the stretch goals, and have a fantastic start producing what I believe will prove to be a game-changing educational product.
But today, I don't want to talk about that. I want to think about multiplication again. Specifically:
Commutativity
I have a memory about commutativity of multiplication. I was in third grade, and I'd been identified as being "good at math." So one day, they moved me up into the fifth grade classroom during math time. I was nervous, being in a new room, but also excited. The lesson was on commutativity of multiplication. The teacher wrote on the board 4 x 7 = 7 x 4. She went on to tell us that this was a rule of multiplication. A general fact.
Then I burst into tears. They put me back in third grade for math for the rest of the year.
I’m not sure why it made me so sad. But as an adult with a deep interest in math education, I continue to have an axe to grind with the commutative property of multiplication. First of all, it’s crazy to introduce it as a self-evident truth. Symbolically, it seems obvious, but order matters with most things. If a = “put on socks” and b = “put on shoes,” then ab ≠ ba. And don’t even think about changing the order of putting on a parachute and jumping out of an airplane.
So order usually matters. But with multiplication it doesn’t. Is that obvious though? Most kids learn about multiplication as repeated addition or skip counting first. Is it obvious that skip counting by 13 seventeen times is the same as skip counting by 17 thirteen times? Starting off, the numbers sure look different.
13, 26, 39, …
17, 34, 51, …
And magically, they just happen to both arrive 221 at the appointed moment. What gives?
Another approach: is it obvious that 13 bags with 17 chocolates in each is the same as 17 bags with 13 chocolates in each? What if the numbers are bigger?
And here’s another question worth considering: which is bigger, 13% of 17, or 17% of 13? Not even sure if commutativity applies anymore, are you?
It takes a little bit of work to even notice how big a problem commutativity is. And it’s worth noticing, and being stuck on it! Because if you never get stuck, you don’t see how elegant the visual is that shows us that, indeed, order doesn’t matter when it comes to multiplication.
And there it is. One picture perfectly describes why 4 x 9 = 9 x 4. I don't need to check that they're both equal to 36. I just need to see that I'm looking at the same array from two different angles!
The leap to see that 13 x 17 = 17 x 13 or ab = ba in general is right there. You could probably sketch the general argument for why commutativity works now.
But something deeper has happened too. Henri Poincaré said that mathematics is the art of giving the same name to different things. We’ve just done that. The same thing (the array), seen at different angles is given different names (4 x 7 or 7 x 4). There is bridge between them that wasn't obvious at first! This is real mathematics. And just to drive the point home, I have a challenge for you. Look at the same array above, and see if you can find the right angle to look at it so you see:
1 + 2 + 3 + 4 + 4 + 4 + 4 + 4 + 4 + 3 + 2 + 1
See it? It’s hiding in plain sight.
I'll even give you a hint. Don't look at it at a right angle ;-)
We're over 200% funded! And we've updated our stretch goals
about 4 years ago
– Tue, May 05, 2020 at 12:09:17 AM
We made it over the 200% mark this weekend. Thanks again to all our wonderful backers!
That means we hit our next stretch goal. We'll now be adding 3 new mini-games in with the deck. That will be some combination of 3 games, explorations, or puzzles that I think are particularly fun and interesting to play. Stay tuned - I'll share them as I decide on exactly what they are!
I've also condensed our stretch goals down to three, to hit at lower funding levels. This gives some good challenges for the campaign, and hopefully some real value added to the box. In effect, if we can double our funding level from right now, we'll provide 12 extra prompts and provocations for the cards over email, 3 additional mini-games inside the app, and an app version of multiplication by heart.
